[1]} first considered the necessary conditions on proxy variables to provably recover the underlying causal effect via direct recovery of the hidden confounder. This work was in turn generalized by [2]}. In their work, it is shown that two types of proxy variables are sufficient to recover the true causal effects without explicitly recovering the confounder. One is an outcome-inducing proxy, which correlates with confounders and causes the outcome, and the other is a treatment-inducing proxy which is caused by confounders and correlates with the treatment. In the flight ticket example, we can use the number of views of the ticket reservation page as the outcome-inducing proxy, and the cost of fuel as the treatment-inducing proxy. Given these proxy variables, [2]} show that the true causal effect can be recovered by solving a Fredholm integral equation, which is referred to as the proxy causal learning (PCL) problem.
